Look at Vieta's Formula

]]>And thanks for taking my "LaTex" goof up off the posts!

Your welcome.

]]>Ah yes! Generating Functions. Like x/(x^2-x-1) generates the Fibonacci sequence if you carry the division on forever past the decimal.

And as an attempt to answer your question, stefy, about introducing multisets vs GFs into math curricula, multisets fit "hand in glove" with basic arithmetic and provide "written manipulatives" for understanding arithmetic. GFs are a bit further down the road.

Hey bobbym, I am trying with this post another attempt at having a signature. We will see if it makes it OK.

And thanks for taking my "LaTex" goof up off the posts! I wasn't sure how to do it. Didn't want to mess anything else up.

]]>Just because they are forced to deal with multisets doesn't mean they are ahead with them. Mathematicians have first used them and probably expanded them in some direction and probably have many applications for them.

]]>What is GF? What comes to mind for me is "Gone Fishing" or "Great Fun" but I doubt that that's it.

I don't know if computer scientists have done much theoretical work in the field, but by the very nature of their programming languages they are forced to deal with multisets extensively. That's what I was trying to convey by "ahead in dealing with multisets."

]]>Computer scientists are way ahead of the mathematicians when it comes to dealing with multisets. For example Visual Studio 12 has a list of 42 functions that operate on multisets. However there seems to be quite an analogy between LaTex/PrettyMath and VisualStudio12/PrettyMultisets. The encoding of Visual Studio 12 is much more complicated than the plain mathematical writing of multisets and their operations. That seems quite analogous to the complicated coding in LaTex vs its output. The basics of multiset theory have been known since the 1960's and maybe awhile longer. The ideas are catching on finally in the mathematical community, but have not been introduced into curricula up through at least high school.

There are some interesting correlations between multisets and prime factors of a number M. For example If M=12 then the multiset of prime factors of M is PM={2,2,3}. The number of distinct subsets of PM is always equal to the number of factors of M (FM={1,2,3,4,6,12} has 6 elements).

Also a number M is a perfect square if and only if the number of distinct subsets of PM is odd.

Also HCF(M,N) = X(PMnPM); LCM (M,N)=X(PMuPN); MxN=X(PM+PN) where

X and x indicate products, n is for intersection and u is for union, + dumps PM and PN together.

The list is much longer than this, but this is enough to give the flavor of the correlation.

Why would they introduce multisets into curriculum if they haven't done so with GF's yet?

I am really amused thaou think there is anything theorethic that computer scientists found before mathematicians. One of the first mentions of multisets in math was by Dedekind in a paper of his in from 1888.

]]>There are some interesting correlations between multisets and prime factors of a number M. For example If M=12 then the multiset of prime factors of M is PM={2,2,3}. The number of distinct subsets of PM is always equal to the number of factors of M (FM={1,2,3,4,6,12} has 6 elements).

Also a number M is a perfect square if and only if the number of distinct subsets of PM is odd.

Also HCF(M,N) = X(PMnPM); LCM (M,N)=X(PMuPN); MxN=X(PM+PN) where

X and x indicate products, n is for intersection and u is for union, + dumps PM and PN together.

The list is much longer than this, but this is enough to give the flavor of the correlation.

Multisets or families as they are sometimes called are used in combinatorics.

]]>But I would include formulas about the other half of set theory --- multisets. Not many folks are aware of multisets. Sets which allow only one copy of each type of element (typesets) form the set theory that corresponds to two valued (T/F) logic. Multisets which have at least two of al tleast one type of object (Eg. {x,x,y,z,z,z,t}, {x,x,y,z}, {x,y,y,y}, etc.) together with typesets forms a bigger set theory in which we have extra operations and relationships. Eg. addition: {x,y,y}+{y,z,z} = {x,y,y,y,z,z}. Similarity: A and B are similar if they have the same types of objects but not necessarily the same number of each type. typeset operator T: T{x,x,y,z,z,z} = {x,y,z} outputs one of each type of object. Likeset operator L: L{x,x,y,z,z,z} = {o,o,o,o,o,o} changes all objects into one generic type object named "o".

Of course we still have union (based on maxima) and intersection (based on minima) and equality of sets. Also there are subsets and differences of sets.

Restricting the types to "o" alone we obtain { }, {o}, {o,o}, {o,o,o}, ... to which we can give the names 0, 1, 2, 3, ... These fit "hand in glove" with basic whole number arithmetic. These sets can be used for "written manipulatives". For example {o,o}+{o,o,o}={o,o,o,o,o} written in terms of their names is 2+3=5. Inequalities come from the subset relationships.

One might have a bit of trouble finding much about these on the internet. Most of the work done on these basic concepts has been done by computer sicentists since their programming languages usually include strings that may or may not have multiple copies of some symbols.

I'd be interested to know what you find out about multisets if you are so inclined.

]]>Surely it's 'Formulae'

It is a Latin word, after all...

I used to say formulae in Chemistry...]]>